National Surveys on Drug Use and Health:

Guide to State Tables and Summary of Small Area Estimation Methodology

Capital S R R is equal to the ratio of two quantities. The numerator is the summation of the product of w sub h h and complete sub h h. The denominator is the summation of the product of w sub h h and eligible sub h h.

Return to Equation A.2-1.

Capital I R R is equal to the ratio of two quantities. The numerator is the summation of the product of w sub i and complete sub i. The denominator is the summation of the product of w sub i and selected sub i.

Return to Equation A.2-2.

Capital O R R is equal to the product of capital S R R and capital I R R.

Return to Equation A.2-3.

The model is given by the following equation: log of pi sub a, i, j, k divided by 1 minus pi sub a, i, j, k is equal to the sum of three terms. The first term is given by x transpose sub a, i, j, k times beta sub a. The second term is eta sub a, i. And the third term is nu sub a, i, j.

Return to Equation B.1-1.

Lower sub s and a is defined as the exponent of capital L sub s and a divided by the sum of 1 and the exponent of capital L sub s and a. And upper sub s and a is defined as the exponent of capital U sub s and a divided by the sum of 1 and the exponent of capital U sub s and a.

Return to Equation B.6-1.

Capital L sub s and a is defined as the difference of two quantities. The first quantity is the natural logarithm of the ratio of Theta sub s and a and 1 minus Theta sub s and a. The second quantity is the product of 1.96 and the square root of MSE sub s and a, which is the mean square error for State-s and age group-a.

Return to Equation B.6-2.

Capital U sub s and a is defined as the sum of two quantities. The first quantity is the natural logarithm of the ratio of Theta sub s and a and 1 minus Theta sub s and a. The second quantity is the product of 1.96 and the square root of MSE sub s and a, which is the mean square error for State-s and age group-a.

Return to Equation B.6-3.

The mean square error, MSE sub s and a, is defined as the sum of two quantities. The first quantity is the square of the difference of two parts. Part 1 is defined as the natural logarithm of the ratio of capital P sub s and a and 1 minus capital P sub s and a. Part 2 is defined as the natural logarithm of the ratio of Theta sub s and a and 1 minus Theta sub s and a. The second quantity is the posterior variance of the natural logarithm of the ratio of capital P sub s and a and 1 minus capital P sub s and a.

Return to Equation B.6-4.

The average annual rate is defined as 100 times quantity q divided by 2. Quantity q is defined as capital X sub 1 divided by the sum of 0.5 times capital X sub 1 plus capital X sub 2.

Return to Equation B.8-1.

The logit of pi hat is equivalent to the logarithm of pi hat divided by the quantity 1 minus pi hat, which is equal to the sum of the following six quantities: negative 5.972664, the product of 0.0873416 and capital X sub k, the product of 0.3385193 and capital X sub w, the product of 1.9552664 and capital X sub s, the product of 1.1267330 and capital X sub m, and the product of 0.1059137 and capital X sub a. or Pi hat is equal to the ratio of two quantities. The numerator is 1. The denominator is 1 plus e raised to the negative value of the sum of the following six quantities: negative 5.972664, the product of 0.0873416 and capital X sub k, the product of 0.3385193 and capital X sub w, the product of 1.9552664 and capital X sub s, the product of 1.1267330 and capital X sub m, and the product of 0.1059137 and capital X sub a.

Return to Equation (1).

The covariance between the natural logarithm of Theta 1 hat and the natural logarithm of Theta 2 hat is equal to the correlation between the natural logarithm of Theta 1 hat and the natural logarithm of Theta 2 hat multiplied by the square root of the product of the variance v of the natural logarithm of Theta 1 hat and the variance v of the natural logarithm of Theta 2 hat.

Return to Equation B.12-1.

Variance v of the natural logarithm of Theta sub i is equal to the square of quantity q. Quantity q is calculated as the difference between capital U sub i and capital L sub i divided by 2 times 1.96, where i takes values 1 and 2.

Return to Equation B.12-2.

Capital U sub 1 is defined as the natural logarithm of the ratio of 0.1556 and 1 minus 0.1556, which is negative 1.6913. Capital L sub 1 is defined as the natural logarithm of the ratio of 0.1098 and 1 minus 0.1098, which is negative 2.0928.

Return to Equation B.12-3.

Capital U sub 2 is defined as the natural logarithm of the ratio of 0.2061 and 1 minus 0.2061, which is negative 1.3486. Capital L sub 2 is defined as the natural logarithm of the ratio of 0.1473 and 1 minus 0.1473, which is negative 1.7559.

Return to Equation B.12-4.

The estimate of the log-odds ratio, lor hat sub a, is defined as the natural logarithm of the ratio of two quantities. The numerator of the ratio is p 2 sub a divided by 1 minus p 2 sub a. The denominator of the ratio is p 1 sub a divided by 1 minus p 1 sub a, where p1 sub a is 0.1310 and p 2 sub a is 0.1747. The estimate lor hat sub a is calculated to be 0.3395.

Return to Equation B.12-5.

The variance v of the natural logarithm of Theta 1 hat is equal to the square of quantity q. Quantity q is calculated as the difference between capital U sub 1 and capital L sub 1 divided by the product of 2 and 1.96. Here, capital U sub 1 is negative 1.6913, and capital L sub 1 is negative 2.0928. Hence, the variance v of the natural logarithm of Theta 1 hat is calculated to be 0.01049.

Return to Equation B.12-6.

The variance v of the natural logarithm of Theta 2 hat is equal to the square of quantity q. Quantity q is calculated as the difference between capital U sub 2 and capital L sub 2 divided by the product of 2 and 1.96. Here, capital U sub 2 is negative 1.3486, and capital L sub 2 is negative 1.7559. Hence, the variance v of the natural logarithm of Theta 2 hat is calculated to be 0.01080.

Return to Equation B.12-7.

Quantity z is the estimate of the log-odds ratio, lor hat sub a, divided by the square root of the sum of the variance v of the natural logarithm of Theta 1 hat and the variance v of the natural logarithm of Theta 2 hat, where lor hat sub a is 0.3395, the variance v of the natural logarithm of Theta 1 hat is 0.01049, and the variance v of the natural logarithm of Theta 2 hat is 0.01080. The statistic z is calculated to be 2.3268.

Return to Equation B.12-8.